Is “The fate of Cai Mao.” true?

The novel describes that Cai Mao was framed by Zhou Yu using a stratagem of sowing discord (Jiang Gan stealing the letter), and Cao Cao mistakenly believed that he was colluding with Wu and had him beheaded (Chapter 45 of Romance of the Three Kingdoms: "Cao was furious... and ordered the warriors to drag him out and behead him."). In official history, it is only recorded that Cai Mao was a supporter of Liu Cong, and followed Liu Cong in surrendering to Cao Cao; the main text does not record that he was wrongly killed by Cao Cao due to the stratagem of sowing discord (Records of the Three Kingdoms, Book of Wei, Biography of Liu Biao: "Liu Biao and his wife loved their younger son Cong, and wanted to make him heir, and Cai Mao and Zhang Yun were their supporters.").